Problem 7 · 1998 AJHSME
Medium
Arithmetic & Operations
regrouping
100 × 19.98 × 1.998 × 1000 =
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Answer: D — (1998)².
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Hint 1 of 2
The answers are all perfect squares, so don't multiply blindly — the four factors are begging to be paired into two EQUAL numbers. Notice every factor is built from the digits 1998.
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Hint 2 of 2
Moving a decimal point is just multiplying or dividing by 10. Slide the points: 100 × 19.98 fattens 19.98 into 1998, and 1.998 × 1000 does the same.
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Approach: pair the factors so the decimal points cancel into 1998 × 1998
- All the choices are squares, which hints the product should split into two identical factors. Pair the factors that 'undo' each other's decimal point: 100 × 19.98 = 1998, and 1.998 × 1000 = 1998.
- So the product is 1998 × 1998 = (1998)².
- Why this transfers: multiplying by a power of 10 only shuffles the decimal point — count how many places each factor moves the point. Pairing factors to cancel awkward decimals beats slogging through a giant multiplication.
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